Question

Under a tension F, it takes 2.00 s for a pulse to travel the length of a taut wire. What tension is required (in terms of F) for the pulse to take 6.00 s instead? Explain how you arrive at your answer.

Short answer

\(\frac{F}{9}\)

Step-by-step solution

Given Data

\(\begin{array}{l}{t_1} = 2\;{\rm{s}}\\{t_2} = 6\;{\rm{s}}\end{array}\)

Concept

The phenomenon is defined as Tension When the forces acting on end of the elements.

Determine the tension

We know that,

\(\begin{array}{c}v = \sqrt {\frac{F}{\mu }} \\t = \frac{L}{v}\\t = \frac{{L\sqrt \mu }}{{\sqrt F }}\\{t_1} = \frac{{L\sqrt \mu }}{{\sqrt {{F_1}} }}\\{t_2} = \frac{{L\sqrt \mu }}{{\sqrt {{F_2}} }}\end{array}\)\(\begin{array}{c}\frac{{{t_2}}}{{{t_1}}} = \sqrt {\frac{{{F_1}}}{{{F_2}}}} \\{F_2} = {\left( {\frac{{{t_1}}}{{{t_2}}}} \right)^2}{F_1}\\ = {\left( {\frac{2}{6}} \right)^2}{F_1}\\ = \frac{F}{9}\end{array}\)

Hence the tension is \(\frac{F}{9}\).